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UC-NRLF 


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IN   MEMORIAM 
FLORIAN  CAJORl 


Heath's  Mathematical  Monographs  | 

r,  ''       Mf-ral  editorship  of  |  - 

Webster  Wells,  S.B. 

:  .;;^.ru-  of  Mathematics   ■-     -c   Massachusetts   Institute   of  Technology 


SUPPLEMENTARY 
ALGEBRA 


BY 


R.  L.  SHORT 


Heath  h  Co.,  Piibhs^^ 


New  York  Chicago 


Number    lo  Price^   Ten  Cents 


The  Ideal  Geometry' 

Must  have 

Clear  and  concise  types  of  formal  dem- 
onstration* 

Many  safeguards  against  illogical  and 
inaccurate  proof. 

Numerous  carefully  graded,  original 
problems. 

Must 

Afford  ample  opportunity  for  origin- 
ality of  statement  and  phraseology 
without  permitting  inaccuracy* 

Call  into  play  the  inventive  powers 
without  opening  the  way  for  loose 
demonstration. 


Wells's  Essentials  of  Geometry 
meets    all   of  these    demands. 


Half  leathery  Plane  and  Solid,  J gQ  pages.  Price  $1.2^^. 
Plane^  'j^  cents.      Solid,  75  ce;ns. 


D.  C.  H  EATH  &  CO.,  Publishers 

BOSTON  NEW  YORK.  CHICAGO 


STUDENTS   IN  ATTENDANCE  AT   STATE   UNIVERSITIES   OF 
THE  CENTRAL   WEST,    1885-1903. 


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Jllinoia 
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Michigan 
Indiana      - 
Kansas      •— 

Minneiota  . 
Ohio 


SUPPLEMENTARY 
ALGEBRA 


BY 

R.   L.  [short 


BOSTON,  U.S.A. 
D.   C.   HEATH   &  CO.,   PUBLISHERS 

190S 


CAJORl 

Copyright,  1905, 
By  D.  C.  Heath  &  Co. 


QA2, 

S55 


PREFACE 

The  large  number  of  requests  coming  from 
teachers  for  supplementary  work  in  algebra,  espe- 
cially such  work  as  cannot  be  profitably  intro- 
duced into  a  text,  has  led  me  to  collect  such 
material  into  a  monograph,  hoping  by  this  means 
to  furnish  the  teacher  with  methods  and  supple- 
mentary work  by  which  he  may  brighten  up  the 
algebra  review. 

As  far  as  possible,  illustrations  have  been  drawn 
directly  from  calculus  and  mechanics,  —  this  being 
especially  true  in  the  problems  for  reduction. 
Almost  without  exception,  such  algebraic  forms 
are  common  to  calculus  work.  For  a  large  part 
of  this  list  of  algebraic  forms  in  calculus,  I  am 
indebted  to  Miss  Marion  B.  White,  Instructor  in 
Mathematics  in  the  University  of  Illinois. 

No  attempt  has  been  made  to  demonstrate 
the  theory  hinted  at  in  graphical  work.  Such 
treatment  would  involve  a  knowledge  of  higher 
mathematics. 

The  graph  of  the  growth  of  state  universities 
was  made  by  members  of  a  freshman  class  of  the 
University  of  Illinois. 

R.  L.  S. 


SUPPLEMENTARY  ALGEBRA. 


Graphs. 


1.  It  is  impossible  to  locate  absolutely  a  point 
in  a  plane.  All  measurements  are  purely  relative, 
and  all  positions  in  a  plane  or  in  space  are  likewise 
relative.  Since  a  plane  is  infinite  in  length  and 
infinite  in  breadth,  it  is  necessary  to  have  some 
fixed  form  from  which  one  can  take  measurements. 
For  this  form,  assumed  fixed  in  a  plane,  Descartes 
(1596-1650)  chose  two  intersecting  lines  as  a  co- 
ordinate system.  Such  a  system  of  coordinates 
has  since  his  time  been  called  Cartesian.  It  will 
best  suit  our  purpose  to  choose  lines  intersecting 
at  right  angles.  ' 

2.  The  Point.  If  we  take  any  point  My  its  posi- 
tion is  determined  by  the  length  of  the  lines  QM= 
X  and  PM  =  y,  the  directions  of  which  are  paral- 
lel to  the  intersecting  lines  C^Xand  (9F(Fig.  i). 
The  values  x  =  a  and  y  =  b  will  thus  determine  a 
point.  The  unit  of  length  can  be  arbitrarily  chosen, 
but  when  once  fixed  remains  the  same  throughout 
the  problem  under  discussion.  QM=x  and  PM 
=y,  we  call  the  coordinates  of  the  point  M.  x, 
measured  parallel  to  the  line  OX^  is  called  the 
abscissa,    y,  measured  parallel  to  the  Hne  OY,  is 

5 


6  Supplementary  Algebra. 

the  ordinate.  OX  and  0  V  are  the  coordinate  axes. 
OX  is  the  axis  of  x^  also  called  the  axis  of  abscissas. 
(9  Fis  the  axis  of  y,  also  called  the  axis  of  ordinates. 
(9,  the  point  of  intersection,  is  called  the  origin. 


Fig.  I. 


A  plane  has  an  infinity  of  points  in  its  length, 
also  an  infinity  of  points  in  its  breadth.  The  num- 
ber of  points  in  a  plane  being  thus,  oo^,  or  twofold 
infinite,  two  measurements  are  necessary  to  locate 
a  point  in  a  plane. 

For  example,  x=2  holds  for  any  point  on  the 
line  AB  (Fig.  2).  But  if  in  addition  we  demand 
that  J  =  3,  the  point  is  fully  determined  by  the 
intersection  of  the  lines  AB  and  CD^  any  point  on 
CD  satisfying  the  equation  j?'  =  3. 

3.  The  Line.  Examine  one  of  the  simplest  con- 
ditions in  X  and  J,  for  example  ;ir  -j- j  =  6.  In  this 
equation,  when  values  are  assigned  to  x^  we  get  a 


Graphs. 


Y 

A 

c 

D 

a' 

X 

X 

0 

Y      ^ 

B 

Fig.  2. 


value  of  J/  for  every  such  value  of  x.  When  x==  o, 
j  =  6;jr=i,j=S;;ir=2,  j  =  4;;ir=3,  j=3; 
;ir  =  5,  J  =  I ;  etc.,  giving  an  infinite  number  of 
values  of  x  and  j  which  satisfy  the  equation.  There 
is,  then,  no  definite  solution. 

Laying  off  these  values  on  a  pair  of  axes,  as 
shown  in  paragraph  2,  we  see  that  the  points  satis- 
fying this  equation  lie  on  the  line  AB  (Fig.  3).  It 
is  readily  seen  that  there  might  be  confusion  as  to 
the  direction  from  the  origin  in  which  the  measure- 
ments should  be  taken.  This  is  avoided  by  a 
simple  convention  in  signs.  Negative  values  of  :r 
are  measured  to  the  left  of  the  7-axis,  positive  to 
the  right.  In  like  manner,  negative  values  of  j/  are 
measured  downward  from  the  ;r-axis,  positive 
values  upward.  The  regions  XO  F,  VOX^,  X'O  Y\ 
Y^OXy  are  spoken  of  as  the  first,  second,  third,  and 
fourth  quadrants  respectively.     (See  Fig.  2.) 


8 


Supplementary  Algebra. 


By  plotting  other  equations  of  the  first  degree  in 
two  variables  (two  unknown  quantities),  it  will  be 
seen   that   such  an  equation  always  represents  a 


Fig.  3. 


straight  line.  This  line  AB  (Fig.  3)  is  called  the 
graph  of  X  +j/  =  6  and  is  the  locus  of  all  the  points 
satisfying  that  equation. 

4.  Now  plot  two  simultaneous  equations  of  the 
first  degree  on  the  same  axes,  e.g.  x-\-y  =  6  and 
2x  —  3^=—  3  (Fig.  4),  and  we  see  at  once  that 
the  coordinates  of  the  point  of  intersection  have  the 
same  values  as  the  x  and  y  of  the  algebraic  solution 
of  the  equations. 

This  is,  then,  a  geometric  or  graphical  reason 
why  there  is  but  one  solution  to  a  pair  of  simulta- 
neous equations  of  the  first  degree  in  two  unknowns. 
A  simple  algebraic  proof  will  be  given  in  the  next 


Graphs,  9 

article.     Hereafter  an  equation  of  the  first  degree 

in  two  variables  will  be   spoken  of  as  a  linear 
equation. 


\ 

Y 

\ 

\ 

D 

\ 

,-/ 

y 

\ 

y 

A 

> 

? 

\ 

/ 

A 

\ 

J/ 

y 

\ 

.^ 

c 

^ 

0 

^5 

J 

Fig.  4. 

5.  Algebraic  Proof  of  Art.  4.  Two  simultaneous 
equations  of  the  first  degree  cannot  have  two  sets 
of  values  for  x  and  y.     Given  the  two  equations 

ax-\-by=^c,  (i) 

ex  -{-fy  =  k,  (2) 

Eliminating  J/,  {af  —  eh)x  =  cf  —  bh,  (3) 

Let  x^  and  x^  be  the  roots  of  (3),  different  in 
value.     Substituting  these  roots,  we  have 

{af—  eb)x^  =  cf—bky 

{of—  eb^x^  =  cf—  bk, 

(of—  eb){x^  —  x^)  -  o. 


lo  Supplementary  Algebra. 

But  x^^x^^y  ,\af=eb,  or  -  =  — ,  which  is  im- 
possible. "^ 

In  general,  the  plotting  of  two  graphs  on  the 
same  axes  will  determine  all  the  real  solutions 
of  the  two  equations,  each  point  of  intersection  of 
the  graphs  corresponding  to  a  value  of  x  and  y 
satisfying  both  equations. 

6.  It  is  well  to  introduce  the  subject  of  graphs 
by  the  use  of  concrete  problems  which  depend  on 
two  conditions  and  which  can  be  solved  without 
mention  of  the  word  equation. 

Prof.  F.  E.  Nipher,  Washington  University,  St. 
Louis,  proposes  the  following : 

"A  person  wishing  a  number  of  copies  of  a  letter 
made,  went  to  a  typewriter  and  learned  that  the 
cost  would  be,  for  mimeograph  work : 
$1.00  for  100  copies, 
1^2.00  for  200  copies, 
$3.00  for  300  copies, 
^.00  for  400  copies,  and  so  on. 

"  He  then  went  to  a  printer  and  was  made  the 
following  terms : 

$2.50  for  100  copies, 
;^3.oo  for  200  copies, 
$3.50  for  300  copies, 

$4.00  for  400  copies,  and  so  on,  a  rise  of  50 
cents  for  each  hundred. 

Plotting  the  data  of  (i)  and  (2)  on  the  same 
axes,  we  have : 


(I) 


(2) 


Graphs 


II 


"The  vertical  axis  being  chosen  for  the  price- 
units,  the  horizontal  axis  for  the  number  of  copies. 

"Any  point  on  line  (i)  will  determine  the  price 
for  a  certain  number  of  mimeograph  copies.  Any 
point  on  line  (2)  determines  the  price  and  corre- 
sponding number  of  copies  of  printer's  work.'* 

Numerous  lessons  can  be  drawn  from  this  prob- 
lem. One  is  that  for  less  than  400  copies,  it  is  less 
expensive  to  patronize  the  mimeographer.  For  400 
copies,  it  does  not  matter  which  party  is  patronized. 
For  no  copies  from  the  mimeographer,  one  pays 
nothing.  How  about  the  cost  of  no  copies  from 
the  printer }     Why  1 

Any  problem  involving  two  related  conditions, 
as  in  line  (i),  depends  on  an  equation. 

The  graph  of  an  equation  will  always  answer 
any  question  one  wishes  to  propound  concerning 
the  conditions  of  the  problem. 

The  graph  offers  an  excellent  scheme  for  the 


12 


Supplementary  Algebra. 


presentation  of  the  solution  of  indeterminate 
equations  for  positive  integers  —  a  subject  always 
hazy  in  the  minds  of  beginners. 

Example.     Solve  3;ir4-47=22  for  positive  inte- 
gers.    Plotting  the  equation,  we  have 


Y 

s 

V 

N 

X 

^ 

s, 

o 

s 

s. 

2 

3 

k 

k 

X        6 

0 

N 

s 

si 

4 
3i 

I 


Fig.  6. 

We  see  that  the  line  crosses  the  corner  of  a 
square  only  when  x=  2  and  x=6.  For  all  other 
integral  values  of  ;r,  y  is  fractional.  The  only  posi- 
tive integral  solutions  are,  therefore,  x=2,  7  =  4; 
x  =  6,  j/=i.  This  corresponds  to  the  algebraic 
result. 

7.  The  Curve.  We  will  now  plot  a  curve  of 
higher  degree  than  the  first.  Take,  for  example, 
;r2  -f  j;/2  =  25,  or  ^2  _-  25  —  ;r2.  Giving  x  either  posi- 
tive or  negative  values,  will  give  positive  and 
negative  values  for  j/.  When  x=±  i,  7  =  ±  V24 ; 
^=±2,  j/  =  ±V2i;  x=±^yj/=±4;  etc.,  until 
•^=  ±  S>  after  which  value  we  find  that  y  becomes 
imaginary.     Plotting  these  values,  we  find  that  our 


Graphs. 


13 


equation  is  represented  by  a  circle  whose  radius  is 
5  (Fig.  7). 

Again,  substituting  values  of  x  in  x^-\-y^^2^y 
we  have  the  sets  of  points  in  Fig.  7,  which,  when 
joined  by  a  smooth  curve,  give  a  circle  of  radius  5 
for  the  locus  of  the  equation, 
r 


X 

y 

±0 

±5 

±1 

±V24 

±2 

±V2T 

±3 

±4 

±4 

±3 

±5 

±0 

Fig.  7. 


2. 


Examples. 

s. 

6. 

3- ^=  I. 

25      16  7. 

It  will  be  noted  that  the  graph  of  every  equation 
in  two  variables  of  higher  degree  than  the  first  is  a 
curve.  Solve  by  means  of  graphs,  and  compare 
results  with  the  algebraic  solution. 


X' 

25      16 

16 


+4=1. 


yl- 


16. 

y  =  x^  —  x—  12. 


14 


Supplementary  Algebra. 


7^ 

JJ 

1 

t 

1-  i  - 

:    t 

V  t  - 

r 

\  t  - 

j^t  -^ 

0 

3 

X 

J' 

-3 

—  20 

—  2 

O 

—  I 

6 

O 

4 

I 

o 

i 

-i 

2 

o 

3 

lO 

Fig.  8. 


Graphs.  ir 

2.    x^+J/^=  13,   ;t:2~j^2_.  j^^ 

It  is  seen  in  Example  i  that  the  line  cuts  the 
curve  in  two  points.  It  is,  in  general,  true  that  a 
straight  line  cuts  a  quadratic  curve  in  two  points 
only.  These  points  may  be  real  and  different,  real 
and  coincident,  or  both  imaginary. 

8.  Examine  the  curve  f  =  x^  —  x^  —  4x  +  4.  Al- 
lowing X  to  vary,  we  obtain  the  set  of  values, 
x=-  3,7=-20;  x=:-2yj/  =  o;  x=-i,j/  =  6; 
;r=o,j/  =  4,  etc.  (Fig.  8). 

Plotting  our  curve,  we  see  that  it  has  something 
of  an  inverted  S-form,  that  it  has  two  bends  or 
turning  points ;  also,  that  it  cuts  the  ;r-axis  in  three 
points;  namely,  when;ir=  i,  ;»r=  —  2,  ;r=  2.  These 
are  the  roots  of  the  equation  just  plotted. 

In  general,  there  are  as  many  roots  to  an  equa- 
tion as  there  are  units  in  the  degree  of  the  curve. 
Now  these  roots  or  intersections  may  not  all  be 
real.  Take,  for  example,  the  curve  j/  =  x^-\-x^  — 
2;ir+  12  (see  Fig.  9). 

We  see  that  the  curve  has  the  same  general  form 
as  the  curve  in  Fig.  6,  but  that  it  crosses  the  axis 
in  only  one  point;  namely,  x=  —3.  An  algebraic 
solution  gives  imaginary  results  for  the  other  two 
roots,  x=  I  +  V—  3  and  x=  i  —  V—  3.  That  is, 
these  two  crossing  points  are  imaginary. 


i6 


Supplementary  Algebra. 


X 

y 

-4 

-28 

-3 

o 

—  2 

12 

—  I 

14 

O 

12 

1 

"M 
"M 

I 

12 

2 

20 

H- 


Fig.  9. 


Graphs. 


17 


This  shows  also,  to  some  extent,  why  imaginary 
points  always  go  in  pairs,  for  it  is  impossible  to 
draw  a  curve  so  that  it  will  cross  the  axis  in  only 
one  imaginary  point,  or,  in  fact,  in  any  odd  number 
of  imaginary  points. 

Imaginary  intersections  may  also  occur  in  simul- 
taneous equations.  In  general,  two  simultaneous 
equations  of  the  second  degree  intersect  in  four 
points,  but  the  curves  may  be  so  situated  that  two, 
or  perhaps  all  four,  of  the  intersections  may  be 
imaginary.     Take  the  curves. 


7- 


Sz 


7- 


k 


-N 


Fig.  10. 


Plotting  these  curves,  we  see  that  the  first  repre- 
sents a  circle  with  its  center  at  the  point  x=  2, 
y  =  6,  the  radius  of  which  is  2 ;  the  second  repre- 


1 8  Supplementary  Algebra. 

sents  a  circle  having  its  center  at  the  origin  and 
a  radius  equal  to  2 ;  also  that  these  circles  do 
not  intersect.     Solving  the  equations  algebraically, 


we  get  for  the  solutions,  x  =  • 


5 


\  both  of 


which  are  imaginary.    (See  Fig.  lo.)    The^-values 
are  also  imaginary. 

If  we  replace  that  first  circle  by 

(^~l)2  +  (j-  1)2  =  4, 

we  get  real  points  for  the  algebraic  solution, 

I  ±  V/           I  T  V/ 
X  = ,  y  =  — 


Fig.  II. 


Plotting  the  curves  (Fig.  ii),  we  see  that  there 
are  also  two  real  intersections. 

Note  that  the  graphical  method  gives  real  inter- 
sections only.     The  reason  is  that  our  axes  are 


Graphs. 


19 


chosen  in  a  plane  of  real  points,  and  imaginary 
points  are  in  an  entirely  different  field.  Imagi- 
naries  are  plotted  by  what  is  known  as  the  Argand 
diagram,  but  are  not  within  the  scope  of  the  present 
paper. 

Example.      Plot }-^=i,   and  x^-\'r^=i6 

25      9 
on  the  same  pair  of  axes.     The  result  should  give 

four  rea/  intersections. 


■ 

■ 

I       '   I    I    I    I 

T' 


Fig.  12. 

9.  The  Absolute  Term.  Determine  what  effect 
the  absolute  term  has  on  the  curve.  Examine  the 
curve  j/=x^—x—i2  (Fig.  12).  The  curve  turns  at 
the  point  x=^y  j/=  —  12^,  and  when;ir=o,  j^=  —  12. 
Replace  12  by  another  constant,  say  6.  The  curve 
now  becomes  j^=x^—x—6  (Fig.  13),  which  turns 


20 


Supplementary  Algebra. 


at  the  point  x=^y  j^  =  —  6|,  and  crosses  the 
j^-axis  at  —6.  Replace  6  by  o.  y^x^-'X  —  o 
(Fig.  14).  This  curve  turns  at  ;r  =  |,  7  =  —  J,  and 
crosses  the  axes  at  the  origin. 


Fig.  14. 


Graphs. 


21 


In  these  three  figures  the  curves  are  of  the  same 
form,  but  are  situated  differently  with  respect  to 
the  axes.  It  is  seen,  then,  that  the  absolute  term 
has  to  do  with  the  position  of  the  curve,  but  has 
nothing  to  do  with  its  form.  Moreover,  a  change 
in  the  absolute  term  lowers  or  raises  the  curve, 
and  does  not  shift  it  to  the  right  or  left. 


1 

i 

t 

1 

\ 

0 

X 

I 
I 

/ 

1 

I 

/ 
1 

\ 

1 

1 

\ 

\ 

1 

1 

\ 

1 

1 

< 

1 

1 

\ 

1 

\ 

\ 

/ 

\ 

/ 

\ 

\J 

J 

X 

1 

' 

V 

V 

r 

■ 

Fig.  15. 

ID.  Change  in  the  Size  of  Roots.  We  learn  in 
the  theory  of  equations  that  the  substitution  of 
x-\-n  for  Xy  where  n  is  positive,  gives  another 
equation,  the  roots  of  which  are  each  less  by  n 
than  the  roots  of  the  original  equation.  Or,  if 
;r  —  ;^  be  substituted  for  ;r,  where  n  is  positive,  each 
root  of  the  new  equation  will  be  greater  by  n  than 


22  Supplementary  Algebra, 

the  roots  of  the  original  equation.  In  4:  =  6,  if 
;r  +  2  is  substituted  for  ;r,  we  get  x=  4,  3.  root  less 
by  2  than  the  former  equation. 

Plot  the  curve  y  =  x^  —  x  —  12  (Fig.  1 5),  the  con- 
tinuous curve.  Substitute  x-\-  2  for  x  in  the  above 
equation,  and  we  have  j/  =  (x+  2j^  —  {x+  2)—  12, 
or7  =  ;r2  +  3;i;—  10;  plot  on  the  same  axes  —  the 
dotted  curve.  The  transformed  curve  crosses  the 
axis  at  x=2f  x==  —  S,  while  the  original  curve 
crosses  at  -;ir  =  4,  ;ir=  —  3,  the  new  roots  each  being 
2  less  than  the  old  ones,  while  the  form  of  the 
curve  is  unchanged. 

We  see,  then,  that  a  substitution  of  x±n  for  x 
shifts  the  curve,  parallel  to  itself,  to  the  right  or 
left,  and  does  not  change  its  form.  The  shape  of 
the  curve  must  therefore  depend  on  the  coefficients 
and  the  exponents  of  x  and  j/. 

Plot  f  =  x^,  y=z(^x—if,  y=={x—2fy  on  the 
same  axes.  It  is  seen  that  the  curves  are  the 
same  curve,  merely  shifted  each  time  to  the  left. 
All  curves  of  the  form  j/  =  (x  —  of  can  be  reduced 
to  the  form  j/  =  x^hy  shifting  the  axes. 

Plot  J/  =  x^,  J/  =  x^,  J/  =  x^,  y  =  ;tr2«,  on  the  same 
axes  where  a  is  positive,  and  note  the  similarity 
(but  not  an  identity)  in  the  form  of  the  curves. 

II.    Plotjv  =  -  for  both  positive  and  negative 

values  of  x.  We  see  that  as  x  grows  smaller,  y 
constantly  increases,  and  that  as  x  approaches 
zero,  y  becomes  infinite.     When  x  is  positive,  the 


Graphs. 


23 


curve  lies  wholly  in  the  first  quadrant,  approaching 
positive  infinity  on  the  ;ir-axes  when  y  is  very  small, 
and  positive  infinity  on  the  ^-axes  when  x  is  very 
small  (Fig.  16).  When  x  is  negative,  the  curve  is 
of  the  same  form  and  in  the  third  quadrant. 

Plot  the  curve  y=^—^'     Note  that  if  a  plane 
x^ 

mirror  were  passed  through  the  ^r-axis  perpendicu- 
lar to  the  j-axis,  the  reflection  of  the  curve  y  ^—^ 

I       ^ 
in  the  mirror  would  be  the  graph  oi  y= -• 


Fig.  16. 

It  has  been  seen  in  paragraph  10  that  the  curves 
yz=ix^^  etc.,  touched  the  ;r-axis,  but  did  not  cross 
it.  These  curves  had  equal  roots  (when  jv  =  o, 
(jt  —  i)2  =  o),  which  give  .r  =  i  or  i.  It  is  in  gen- 
eral true  that  a  curve  with  equal  roots  must  be 
tangent  to  the  axis,  i,e,  must  cut  the  axis  in  at 


24  Supplementary  Algebra. 

least  two  ^consecutive  points.  If  the  equation  has 
in  its  numerator  a  binomial  factor  (x  —  a)  repeated 
an  even  number  of  times,  the  curve  will  touch  the 
axis  at  a,  but  will  not  cross  at  that  point.  But  if 
the  factor  be  repeated  an  odd  number  of  times, 
the  curve  will  touch  the  axis  and  cross  it  also  at  a 
[try  J  =  (-^  —  I  /] .  If  the  binomial  is  not  repeated, 
the  curve  crosses  the  axis  at  a  non-vanishing  angle. 
All  curves  where  the  factor  repeats  are  parallel  to 
the  axis  at  the  point  of  intersection.  This  fact 
is  at  once  apparent  when  we  remember  that  the 
curve  is  a  tangent  to  the  axis  at  such  a  point. 

12.  Maxima  and  Minima.  It  may  be  noticed 
that  many  of  these  curves  make  a  complete  turn, 
as  in  Fig.  8  or  Fig.  1 2 ;  that  is,  there  is  some  point 
which  is  higher  or  lower  than  any  other  point  of 
the  curve  in  the  immediate  vicinity  of  the  point  in 
question.  Such  turning  points  are  called  maximum 
and  minimum  points  of  the  function.  These  points 
are  of  considerable  geometric  value. 

Problem.  Suppose  it  be  required  to  find  the 
rectangle  of  greatest  area  which  can  be  inscribed  in 
an  isosceles  triangle  with  an  altitude  equal  to  2  and 
a  base  equal  to  2.  Let  ABE  (Fig.  17)  be  the  given 
triangle,  DCy  one  half  the  required  rectangle.  Let 
J  be  the  area  of  the  rectangle  and  2  x  the  base. 

*  Mathematically  the  word  coincident  should  replace  consecutive. 
The  beginner,  however,  seems  more  clearly  to  realize  that  the 
intersections  are  approaching  each  other  indefinitely  near  if  consec- 
utive is  used. 


Graphs. 


25 


~"~~ 

X 

0] 

Y 

\       1 

Fig.  18. 

From  the  figure, 

CO  :  AD-OD  : :  2  :  i, 

C0^2{AD-0D\ 

=  2(i-;r), 

therefore  y^^x  —  A^x^  (Fig.  18).  But  this  ex- 
pression is  a  curve  where  y  represents  the  area  we 
are  trying  to  find,  and  may  be  plotted  as  any  other 
curve. 

Plotting  the  curve,  we  see  that  it  turns  at  ;r  =  |-, 
j^=  I  (Fig.  18),  and  that  this  point  is  higher  than 
any  other  point  of  the  curve. 

Therefore,  the  greatest  rectangle  which  can  be 
inscribed  in  the  given  triangle  is  one  in  which  the 
base  and  altitude  are  each  i.  The  rectangle  is 
minimum  when  x=o  and  when  x=  i.     Why  ? 

Again :  Given  a  square  piece  of  sheet  metal  30 
inches  on  a  side,  find  the  side  of  a  square  to  be 


26 


Supplementary  Algebra. 


cut  out  of  each  corner  of  the  sheet  so  that  the 
remainder  will  fold  up  into  a  box  of  maximum 
volume. 

Let  ABCD  (Fig.  19)  be  the  sheet  of  metal  and 
y  the  volume  of  the  box.  Let  x  be  the  side  of  the 
square  to  be  cut  from  each  corner. 


s 

7 

H^ 

Fig.  19. 


Fig.  20. 


Then  from  the  figure,  y  =  (30  —  2  xfx. 
Plotting  this  curve  (Fig.  20)  we  see 
that  it  turns  twice,  once  at  ;r  =  5  and 
once  at  x=  1$;  also  that  ;r=  15  is  at 
the  lowest  part  of  a  bend,  while  ;r=  5  is 
at  the  highest  part  of  the  other  bend. 
This  shows,  then,  that  the  box  is  greatest 
when  ;r=  5  and  least  when  ;r=  15,  for 
when  15  is  taken  from  each  corner  no 
material  remains  from  which  to  make 
the  box.     The  graph  shows  also  that 


X 

I 

0 

0 

2 

1352 

5 

2000 

6 

1944 

8 

1568 

10 

1000 

13 

208 

14 

56 

15 

0 

16 

64 

Graphs.  27 

the   greatest   box   that  can  be  so  formed   has  a 
volume  of  2000,  the  value  of  y  of  the  curve,  when 

The  turning  points  of  a  curve  are  often  difficult 
to  plot  because  the  values  of  x  and  y  must  be  taken 
so  closely  together.  Much  labor  may  often  be 
saved  by  the  following  process,  called  differentia- 
tion. Only  one  rule  is  given  here.  Should  the 
reader  wish  rules  governing  all  cases,  he  will  find 
them  in  Wells's  **  College  Algebra,"  p.  472,  and 
Wells's  "Advanced  Course  in  Algebra,"  p.  527. 

The  following  rule  holds  where j/  equals  a  rational 
polynomial  in  Xy  containing  no  fractions  with  vari- 
ables in  the  denominator :  multiply  each  coefficient 
by  the  exponent  of  x  in  that  term  and  depress  the 
exponent  by  one.  If  the  resulting  expression  is 
equated  to  zero  and  solved  for  x,  the  roots  thus 
obtained  will  be  the  abscissas  of  the  points  where 
the  curve  turns.  A  proof  of  this  will  have  to  be 
postponed  until  higher  mathematics  is  reached. 

Illustration.     Use  the  curve  in  Fig.  8. 

j^  =  ;r3-;tr2-4;r  +  4.  (l) 

Differentiating  and  equating  the  result  to  zero, 
we  have  ^:^-2x~4  =  0,  (2) 

whence     x=— — ^=  1.5+  or  —  ,^6+y 

3 
the  turning  points  of  the  curve.     This  result  may 

be  verified  by  examining   Fig.    18.      These  two 


28 


Supplementary  Algebra. 


Fig.  21. 


Graphs.  29 

curves  are  plotted  on  the  same  axes  in  Fig.  21, 
equation  (2)  being  dotted. 

Problem.  An  open  vessel  is  to  be  constructed 
in  the  form  of  a  rectangular  parallelopiped  with  a 
square  base,  capable  of  containing  4  cubic  inches. 
What  must  be  the  dimensions  to  require  the  least 
amount  of  material  ?     Solve  by  plotting. 

Ans.     Dimensions  must  be  2  x  2  x  i  inches. 

It  is  often  necessary  to  plot  a  curve  in  one  vari- 
able only;  for  example,  ;ir2—;r— 12  =  o.  In  such 
a  case  we  are  really  seeking  the  intersection  of  two 
curves,  j/  =  o  and x^—x—  12  =  0,  which  are  plotted 
as  above,  y  =  o  being  the  Hne  OX.  In  such  a  case 
the  curve  is  readily  plotted  if  the  two  equations  are 
equated ;  e,g,  y^x'^—x—  12. 

Flot  x^  —  X  —  6  =  o;  also  regard  the  equation  as 
of  the  form  ax^  -\-  bx  -{-  c  =  o,  and   solve   by  the 

formula,  x  = .    Note  the  nature  of 

2a 

the  radical  part  of  the  formula  after  the  substitution 
is  made. 

Treat  x^—  x-{-^==o  and  x^—x-{-6  =  o  in  the 
same  manner.  Remember  that  the  places  where 
the  curve  crosses  the  ;ir-axis  correspond  exactly  with 
the  algebraic  root  obtained.  This  affords  an  ex- 
cellent method  for  illustrating  to  the  student  the 
meaning  of  imaginary  roots. 


30  Supplementary  Algebra. 

Short  Methods. 

(To  shorten  the  work  in  certain  classes  of  fractions  involving 
addition.     For  review  work  only.) 

Rule  I.  If  two  fractional  numbers  having  a 
common  numerator  and  denominators  prime  to 
each  other  are  to  be  added,  multiply  the  sum  of 
the  denominators  by  the  common  numerator  for 
the  numerator  of  the  result;  the  product  of  the 
denominators  will  be  the  denominator  of  the  result. 

Rule  II.  If  two  fractional  numbers  having  a 
common  numerator  and  the  denominators  prime 
to  each  other  are  to  be  subtracted,  subtract  the 
first  denominator  from  the  second,  multiply  this 
difference  of  the  denominators  by  the  common 
numerator  for  the  numerator  of  the  result;  the 
product  of  the  denominators  will  be  the  denomi- 
nator of  the  result. 

Illustration  : 

i  +  *  =  T^;  f  +  l  =  2G  +  i)  =  if- 
4-i=A;  (f-l)  =  2(J-i)  =  ^5- 

This  scheme  makes  possible  a  great  saving  of 
products  in  algebraic  summations. 


+  ■ 


(use  Rule  II), 


Short  Methods.  31 

_     4      ,     -4  ^J     I i_\ 

(use  Rule  II), 
12 


-"(^4-4X^4-1) 

2.  — "^ ^=-J! —'   (Use  Rule  II.) 

X—l       X—2       ^—3       X—4 

—  I I . 

(x-i){x-2)'~'^{x-3)(x-4y 
whence 

(^-3)(^-4)  =  (-^-"i)(^-2); 
:ir2— 7;ir+  12  =  ;ir2  —  3;ir4- 2. 
.-.  ;r=2l 

3.  7=  H 7==  =  12  :  solve  for  x. 

X  +  V ;r2  —  I      X  —  -Vx^  —  I 

(     ,    /% + 7t=V  ^2  (use  Rule  I); 

\x  +  V;!:^  —  I      X  —  -yx"^  —  1/ 

24r 

12. 


;r2  — (;r2—  i) 

Improper  Fractions. 

The  larger  the  factor,  the  more  the  multiplication 
involved,  the  greater  the  liability  of  error,  and  the 
slower  the  speed.  The  times  demand,  in  mathe- 
matical computations,  accuracy  combined  with  a 
reasonable  degree  of  rapidity. 


32  Supplementary  Algebra. 

In  general,  improper  fractions  have  no  place  in 
problems  involving  addition  and  subtraction. 

In  arithmetic  we  do  not  add  |,  f ,  f ,  etc.,  because 
the  work  is  cumbersome  and  the  factors  large. 
We  reduce  these  fractions  to  whole  or  mixed  num- 
bers, 2f,  2 J,  if,  add  the  integers,  and  then  the 
fractions.  We  should  not  think  of  adding  the 
fractions  in  their  original  form.  The  same  prac- 
tice holds  good  for  algebra.  An  improper  fraction 
may  be  defined  as  a  fraction  in  which  the  numer- 
ator is  of  the  same  or  higher  degree  than  the 
denominator,  both  being  rational. 

Examples. 

X—l       x— 2  _x  —  3       X  —  ^ 
X—2      X  —  2i      ^—4      ^-"5 
Divide  each  numerator  by  its  denominator  — 
parenthesis  division. 

I                    I.I  I 

H I -.=  iH ^- i_. 


X--2  x—i  ^—4  ^—5 

Collecting, 

\_X-2       X-l\       \_X-\       X-<>\ 

(Use  Rule  II.) 

—I        _  —  I 

(Ar-2)(;r-3)~(;tr-4)(^-5)' 
{x-^)(x-l)  =  (x-2){x-z). 


Short  Methods.  ^3 

Such  reduction  of  improper  fractions  will  in  most 
cases  shorten  a  solution  and  will  reduce  the  size  of 
the  factors ;  a  binomial  numerator  generally  reduces 
to  a  monomial,  and  a  trinomial  numerator  to  a 
binomial. 

2.         ^        ^    ^ 


^  +  7^+  lO       X  —  2 

x^-{- yx-\-  lo 


Reducing,  i  +   o^,^^,^ -  i 


Collecting,  factoring  first  denominator, 

7 r-r^^ r bCCOmOS      ; ]• 

{x -{- i){x -\- 2)       X—2  \x-\-2       X—2j' 

(Use  Rule  II.) 

X^—X-^-  I    .   X^-\-X'\-  I 

3. 1 ; ^2X. 

X—  I  X-\'  I 

Reducing,    x-^- — 7  +  -^  +  rT:7  =  2;r, 
X  —  1  X  "i"  1 

whence, 1 ; —  =  o.  (Apply  Rule  I.) 

X—l       X-\-l  \     irtr  J  / 

r.x  —  o. 

Problems  of  the  above  type  are  common  to  all 
texts  on  algebra.  After  the  student  has  solved 
them  by  the  usual  methods,  so  as  to  become  fa- 
miliar with  the  principles  involved,  he  should  be 
given  a  review  where  these  briefer  solutions  might 
be  introduced. 


34  Supplementary  Algebra. 

Note.  A  short  method  should  never  be  introduced  early 
in  a  subject.  Such  procedure  clouds  the  pupil's  ideas  of  mathe- 
matical principles. 

Simultaneous  Quadratics  of  Homogeneous  Form 
may  often  be  solved  as  follows : 

Illustrative   Problems. 


Eliminate  the  abso- 
lute terms  by  multi- 
plying equation  ( i )  by 
(c)  9f  equation  (2)  by  14, 
and  subtracting. 

This  resulting  func- 
tion of  (;r,^)is  always 
capable  of  being  fac- 
tored, either  by  in- 
spection or  by  quad- 
ratic factoring  and 
gives  two  values  of  x 
in  terms  of  j.  These, 
when  substituted  in 
( I )  or  (2)  will  produce 
the  required  values 
for  J/.  X  is  then 
easily  obtained.  This 
method  avoids  the 
substitution  of  y'=^vx. 


2x^  —  xy^2S 

;i:2  +  2/=  18 

(I) 
(2) 

i%x^  —  gxy  =  2S2 

l4;r2  +  2872=252 

(3) 
(4) 

4;r2  —  ()xy  —  2872  =  0 

(5) 

(4^+7j/)(^~4J^)  = 

0. 

,\x=-lyor  4y. 

Substituting  in  (2), 

11/ +  2/ =18. 

^^^^-18. 

16 

y'=¥- 

.•.J/=±|V2. 

16/ +2/=  18. 

y=i. 

±  I. 


Short  Methods.  35 

Algebraic  Problems. 

The  following  problems  in  reduction  are  of  com- 
mon occurrence  in  the  higher  mathematics,  and 
especially  in  the  calculus,  where  long  lists  of  such 
forms  are  found  in  the  work  almost  daily.  They 
appear  most  frequently  in  differentiation  and  prep- 
aration for  integration  ;  lack  of  facility  in  handling 
such  forms  handicaps  a  pupil  and  makes  him  think 
the  calculus  a  difficult  subject. 

Type  Forms. 

(a  +  df  -{-(a-df=2  (a^  +  6^)  ; 

{a  +  df-4ad  =  (a'-df; 
{a-df-h4ab  =  {a  +  df. 

It  is  desirable  that  a  student  be  so  familiar  with 
these  type  forms  that  he  may,  by  inspection,  write 
down  the  results  of  similar  operations. 

a  -]-  d  a  —  b 


(7  +  V25  -/)2  -  (7  -  V25  -/)2  =  .? 

Simplify  the  following : 
{e^'he-^f-^4, 


^6  Supplementary  Algebra. 

■\/(mn  —  i)2  +  4  mfif 

Negative  Exponents. 

Notice  all  quantities  which  have  negative  expo- 
nents. Multiply  both  numerator  and  denominator 
of  the  given  expression  by  the  product  of  all  such 
quantities  with  the  sign  of  the  exponents  changed. 
If  the  expression  be  integral,  regard  its  denomina- 
tor as  unity. 

Example  (i): 

Multiply  both  numerator  and  denominator  by 

2 

Result  =  ^  ^ 


Example  (2) : 

x(i+x^y^-x{i-x^y^. 


•y/l  — X^\I  -\- X^ 

Multiplying  by  ,  we  obtain 

Vl  —x'^^Y  ■\-X^ 
x^i  —x'^  —  x^i  '\-X^ 


Short  Methods.  37 


In  like  manner  simplify : 


,-.+^^  fif+f-^-* 


jrJj/* 


In  all  reductions  watch  for  opportunities  to 
factor,  especially  to  remove  factors  common  to 
several  terms. 

(«)  Z{x -  2f{2 X -^  if -^  t,{2  X ^  \)\x -  2f 

=  {2X+  lf(x-2f(l8x-  II). 

(b)  3{x-i)\x  +  2f  +  4(x-if{x+2f 

=  {x-if{x  +  2f(7x+s). 

(c)  (a^+x^f-  3  x^(a^+  x^f  =  y/'^V^{a^-  2  :fi). 
(^)  8;r(4;r2  -  3)(;r2  +  l)*+ 24;r(;^  +  I)* 

=  56;t8(;tr2+i)*. 

(^)   |(;r+l)-^(;p-5)2  +  2(x-5)(;ir+l)* 

=  |(;r+l)-i(^-5)(2;r-l). 

Problems. 


1.   Reduce    == to  the  form 


Vi-;.^  i-^ 


38  Supplementary  Algebra. 


a 


2.  Reduce    s-f^^ -- 

a^        x-\-a 

x^       x  —  a 

3.  Reduce 


4.    Reduce 

I 


V^ 


to  the  form 


ax  —  x'' 


■H^-i^' 


5.    Reduce 

m  _m  m  jm 

to  the  form 


X'^—X    ""  x^-\-x~^ 

6.    Reduce 


'  +  ■\/x^  —  cfi      x^  jx^  —  a 


a  ^      d^ 


to  the  form  -\ 

X  ^  x  —  a 


7.    Reduce 
to  unity. 


Short  Methods. 


39 


8.   Reduce 
ax 


{a^  -  x^)i 


to  the  form 


V^2z:^ 


9.    S^VWK\        If    K^^^''y-y\    find 
/ —  -^ 

10.  Reduce  2 
I 

Vl  —  3;ir  — ;tr2 

11.  Reduce 


to  the  form 


V13 


:x^-x^-6 


to  the  form  — 


—  4;^^ 

25  I  A-2^y 


12.    Reduce  V2  ax  —  ;r2  to  the  form 
a {x  —  df 


13.   2  tan  ;ir  4-(tan  ;i:)2  —  3  =  o ;  find  tan ;ir. 


14.    2  cos  ;r  H =  3  ;  find  cos  x. 

cos;r 


40  Supplementary  Algebra. 


15. h2  cot;r=fVi -f-cot2;r;  find  cot ;r. 

cot;r 

16.  Reduce  ; r-^ to  the  form 

L_  + I 4 

2(l+-sr)       I0(l--8r)       5(3  4-  2-8') 

17.  vS  =  — ^(i2  +  xy  -  8.      Evaluate  5  when 

18.  Reduce 


M     4^      V 

V2V1-W 


2x—  2;rVi  —x^ 


to  the  form        -li-\ —  V 

19.    Reduce 

_  .       2;r—  I 


■y/x^  —x—  I 


to  the  form 


2;r—  I  +  2  ^x^—x—  I 

I 


V^^^-jF— I 
20.    Reduce 


/£~2\i  _^  3  fx^2\'\f      2     Y 

V;y+2/        4V;tr+2/     \:rH-2/ 

W+2/ 


+  2/ 
;i^  —  I 

to  the  form  —^ 


Short  Methods. 

21.    Reduce 

[-(^)i^^ 

X            _x 

ea  +  e  a 

2a 

to  the  form              «(^^-^"^X 

4 

22.   Reduce 

—  r 
f 

to  the  form  —y. 

23.   Reduce 

X :i— 

-4/ 

,2 

41 


y 


to  3-^  +  2/  when  y  =  4/;r. 

24.    Reduce 


to 


-<=m 


a^L 


jr2       -1/2 
when  ±~-i^2L=i, 


42  Supplementary  Algebra. 

25.    Reduce 

V       «VA  ah  I 


^ --^ 


26.    Reduce 


^-T/i/  3  /       i/*\        'i/^;tr'~^ 


to 


^s  3  ;r^^ 


when  x^  -hj/^  =  ^^. 


27.    Reduce 

„i\2ll 


1 to  — ^^ — Y^  when  x^  -{-y^  =  a^. 


£^ 

2X 

28.    Reduce 


2a 


1/2  ^     ?  _? 

to  —  when  ^=:-(^  +  ^  «). 


Short  Methods.  43 

29.    Reduce 


to 


2V;r 

{2  a  — xf 

'y/x{2a  —  xf 
30.    Reduce 


31.   ^-^v^:=^=^^l-"^V-^i),      (I) 


^  = r— ^-^^  (2) 


^1 


^1 

Eliminate  ;irj  between  (i)  and  (2)  and  get 
x'^^-y^^ia-Vbf. 

32.    Reduce 

\^x-a^x-{-a  —  \{x'\-df{x  —  aJ^ 
X  —  a 

(x—  2  (i\\/x  4-  d 
to  ^^ — 3 

{x-^df 


44  Supplementary  Algebra. 

33.    Reduce 

n(i'\'xY  *  x""-^  -nii-V  xY-^  •  x'^  nx"^"^ 
^              to 


34.  Reduce 

2(i-a^)-{-4x^ 

35.  Reduce    ^ ^-r^r-  to 


-(tST     ■""■ 


36.    Reduce 


37.    Reduce 
;.(l+^)-^-;t-(l-^)-^  to  ifi  -        ^       1. 


38.   Reduce 


(l-2;ir'^)^         ^    If       4;tr-2 


j^,       -"       ,^        2L2^-2;tr+I 


Vl  -  2  W 

—  4^±?— Ito-^. 
2;r2-h2;r+iJ         1+4^ 


Short  Methods. 


45 


39.    Given 


show  that  (a+/9)*+(a~/3)*=2>^*. 


40.   Given 

a^     (k-af 
a^     {k-af 


;  show  that  x^  +7'  =  /^^ 


Wells's   Mathematical   Series* 

ALGEBRA. 
Wells's  Essentials  of  Algebra      .  •  •  .  ,    $z.zo 

A  new  Algebra  for  secondary  schools.  The  method  of  presenting  the  fundamen- 
tal topics  is  more  logical  than  that  usually  followed.  The  superiority  of  the 
book  also  appears  in  its  definitions,  in  the  demonstrations  and  proofs  of  gen- 
eral laws,  m  the  arrangement  of  topics,  and  in  its  abundance  of  examples. 

Wells's  New  Higher  Algebra      .  .  .  .  .1.3a 

The  first  part  of  this  book  is  identical  with  the  author's  Essentials  of  Algebra. 
To  this  there  are  added  chapters  upon  advanced  topics  adequate  in  scope  and 
difficulty  to  meet  the  maximum  requirement  in  elementary  algebra. 

Wells's  Academic  Algebra  .  .  .  •  ,       x.o8 

This  popular  Algebra  contains  an  abundance  of  carefully  selected  problems. 

W^ells's  Higher  Algebra    ......       1.32 

The  first  half  of  this  book  is  identical  with  the  corresponding  pages  of  the  Aca- 
demic Algebra.    The  latter  half  treats  more  advanced  topics. 

Wells's  College  Algebra   ......       1.50 

Part  II,  beginning  with  Quadratic  Equations,  bound  separately.    $1.32. 

Wells's  Advanced  Course  in  Algebra         ....    $1.5^ 

A  modern  and  rigorous  text-book  for  colleges  and  scientific  schools.  This  is  the 
latest  and  most  advanced  book  in  the  Wells's  series  of  Algebra. 

Wells's  University  Algebra  .  .  •  .  .1.3a 

GEOMETRY. 

Wells's  Essentials  of  Geometry  —  Plane,  75  cts.;  Solid,  75  cts.; 

Plane  and  Solid    .  .  .  .  .  .  .       z.25 

This  new  text  offers  a  practical  combination  of  more  desirable  qualities  than 
any  other  Geometry  ever  published. 

Wells's  Stereoscopic  Views  of  Solid  Geometry  Figures        •        .60 

Ninety-six  cards  in  manila  case. 
Wells's  Elements  of  Geometry  —  Revised  1894.  —  Plane,  75  cts.; 

Solid,  75  cts.;  Plane  and  Solid     .  .  •  •  •       z.as 

TRIGONOMETRY. 
W^ells's  New  Plane  and  Spherical  Trigonometry  •  .     $1.00 

For  colleges  and  technical  schools.    With  Wells's  New  Six-Place  Tables,  $1.25. 

Wells's  Complete  Trigonometry     .  .  .  ,  .        .90 

Plane  and  Spherical.  The  chapters  on  plane  Trigonometry  are  identical  with 
those  of  the  book  described  above.     With  Tables,  $1.08. 

Wells's  New  Plane  Trigonometry               •            •  •            •        .60 

Being  Chapters  I-VIII  of  Wells's  Complete  Trigonometry.  With  Tables,  75  cts. 

Wells's  New  Six-Place  Logarithmic  Tables  •  •  •  .60 
The  handsomest  tables  in  print.    Large  Page. 

Wells's  Four-Place  Tables           .            •            •  •            •        .35 

ARITHMETIC. 
Wells's  Academic  Arithmetic      •  •  •  •  •     $1.00 

Correspondence  regarding  terms  for  introduction 
and  exchange  is  cordially  invited. 

D.  C.  Heath  &  Co.,  Publishers,  Boston,  New  York,  Chicago 


Wells's 
Essentials  of  Geometry 

DEVELOPS    SKILL    IN 

THE   BEST    METHOD    OF    ATTACK 

IN   THE    FOLLOWING    WAYS 


It  shows  the  piipii  at  once  the  parts  of  a  proposition  and  the 
nature  of  a  proof.     Sections  36,  39,  40. 

11. 

It  iat-'-n  'yrv-  ■■■n ;■;;.;.•::!  ::,■.;<;;-,;;,-,  "  •^,.:■H',  and  gives  such  aid  as 
will  lead  the  student  to  see  how  each  proof  depends  upoa  some- 
thing already  given.     See  pages  17-18. 

HI. 

It  begins  very  early  to  leave  something  in  the  proof  for  the 
student  to  do.     See  section  51  and  throughout  the  book. 

The  converse  propositions  are  generally  left  to  the  student.  The 
'-'  indirect  method  ''  is  used  in  those  first  proved  and  thus  gives  a  hint 
as  to  1k)\v  such  propositions  are  disposed  of.     See  page  21. 

V. 

.^  Uj:^o  :i;:iiu.cr  of  the  simpler  propositions,  even  iu  the  first  book, 
are  left  to  the  student  with  but  a  hint.     See  pages  50,  51,  52,  etc. 

VL 
iiguics  with  all  auxiliary  lines  are  made  ilu   e  •  i^ 

the  student  learns  why  and  how  they  are  drawn. 

VIL 
Atter  Book  I  the  authority  for  a  statement  m  the  proot  is  r<ot 
stated,  but  the  quesdon  mark  used  or  the  section  given. 


D.  C.  HEATH  &  CO.,  Publishers 

BOSTON  NEW  YORK  CHICAGO 


MATHEMATICAL  MONOGRAPHS 

ir:SU}-I>  UNDER  THE  GENERAL  EDITORSHIP  OF 

WEBSTER  WELLS,  S.B. 

Profiwmr  of  Matluyncdl  a  in  the  ^fai::■'■■■  /  ;/.:  :'■.;  Insiiiide  of  Technology. 


It  is  the  purpose  of  this  series  to  make  direct  contribution 
to  the  resources  of  teachers  of  mathematics,  by  presenting 
freshly  written  and  interesting  monographs  upon  the  history, 
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1.  FAMOUS   GEOMETRICAL  THEOREMS  AND  PROBLEMS   AND 

THEIR  HISTORY.    By  William  W.  Rupert,  C.E. 

u    TJie  Greek  Geometers,    iu    The  Pythagorean  Proposition. 

2.  FAMOUS  GEOMETRICAL  THEOREMS.     By  William  W.  Rupert. 

ii.    The  Pythagorean  Proposition  (concluded),    iii.     Squaring  the  Circle.' 

3.  FAMOUS  GEOMETRICAL  THEOREMS.    By  W^illlam  W.  RLtptRT. 

jv.     TnsrctK  vn  vA  au  Angle.     Y.     The  Area  of  a  Triangle  in  Ternis  of  its  Sides. 

4.  FAMOUS  GEOMETRICAL  THEOREMS.    By  William  W,  Rupert. 

vj.    The  Duplication   of   the  Cube.      vii.    Mathematical.  Inscription   upon   the 

'i"i>i.i.  u.,;r:  of  LiKlolph  Van  Ccnlcn. 

5.  OJ^    TEACHING  GEOMETRY.     By  Florence  Milner. 

6.  GRAPHS.     By  rrofr-ssor  R.  J.  Ally,  Indiana  University. 

7.  FACXORIK^G.     By  l^ofessor  W,i^:b.ster  Wells. 

PRICE,  lo  CENTS  EACH 


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__~^^j^ojectto  immediate  recall. 


"PR  5   msm 


27Nov'57GC 


lOV  IS  flE7 

22fab'60J(J 
REC'D  CD 

2l-100m-l,'54(1887sl6) 


476 


